Taylor and Maclaurin Series
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Taylor and Maclaurin Series
If we represent some function f(x) as a power series in (x-a), then
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Uniqueness Theorem
Suppose
for every x in some interval around a.
Then
.
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Taylor's Formula with Remainder
Let f(x) be a function such that f(n+1)(x) exists for all x on an open interval
containing a.
Then, for every x in the interval,
where Rn(x) is the remainder (or error).
Taylor's Theorem
Let f be a function with all derivatives in (a-r,a+r).
The Taylor Series
represents f(x) on (a-r,a+r)
if and only if
.
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EX 1 Find the Maclaurin series for f(x)=cos x and prove it represents
cos x for all x.
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EX 2 Find the Maclaurin series for f(x) = sin x.
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EX 3 Write the Taylor series for
centered at a=1.
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EX 4
Find the Taylor series for f(x) = sin x in (x-π/4).
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EX 5 Use what we already know to write a Maclaurin series (5 terms)
for
.
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